Задание 3 · Стереометрия (краткий ответ)
3.4. Площадь поверхности составного многогранника
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1. Базовые формулы
| Тело | Площадь поверхности |
|---|---|
| Прямоугольный параллелепипед с рёбрами \(a,b,c\) | \(S=2(ab+bc+ac)\): каждая из трёх пар противоположных граней даёт своё слагаемое |
| Куб с ребром \(a\) | \(S=6a^2\) |
| Прямая призма | \(S_{\text{бок}}=P_{\text{осн}}\cdot h\), \(S_{\text{полн}}=S_{\text{бок}}+2S_{\text{осн}}\) |
2. Три способа найти площадь поверхности составного многогранника
Составной многогранник (все двугранные углы прямые) состоит из нескольких прямоугольных частей. Выбирайте способ по виду фигуры.
Способ 1. Фигура — прямая призма
Многие составные фигуры можно рассматривать как прямую призму, лежащую на боку, с «ломаным» многоугольником в основании. Тогда
\(S_{\text{полн}}=P_{\text{осн}}\cdot h+2S_{\text{осн}}\).
Площадь основания находят, разбивая его на прямоугольники; периметр — по контуру основания. (Например, при \(P_{\text{осн}}=28\), \(S_{\text{осн}}=44\), \(h=12\) получается \(28\cdot12+2\cdot44=424\).)
Способ 2. Охватывающий параллелепипед и поправка
- Найдите площадь поверхности большого прямоугольного параллелепипеда, в который «вписана» фигура (из него как будто вырезана часть).
- Вычтите площади граней, которых нет у исходной фигуры, и прибавьте площади новых граней, появившихся на месте выреза (стенки выемки).
Способ 3. Сложить площади всех граней
Если фигуру нельзя представить призмой, просто сложите площади всех граней: каждую видимую и невидимую на рисунке, но лежащую на поверхности.
3. Составная фигура как объединение тел
- Разбейте фигуру на простые тела (параллелепипеды, кубы), посчитайте поверхность каждого и сложите.
- Если два тела прилегают друг к другу, их общая грань есть в поверхности каждого, но на поверхности составной фигуры её нет. Поэтому площадь общей грани вычитают дважды — по одному разу для каждого тела.
- Если к телу пристроен выступ, добавляют площадь открытых граней выступа. Если вырезана выемка, вычитают площадь вырезанной грани и учитывают новые стенки выемки.
4. Фигуры из одинаковых кубиков
У каждого кубика 6 граней. Каждая грань, прилегающая к соседнему кубику, на поверхность не выходит. Площадь поверхности — это число открытых граней, умноженное на площадь одной грани. Пример: кубик, окружённый шестью соседями, не даёт открытых граней; каждый из этих шести даёт по 5 открытых граней.
5. Масштабирование
При увеличении всех линейных размеров в \(k\) раз площадь поверхности увеличивается в \(k^2\) раз.
- Вычитают площадь выреза, но забывают учесть новые внутренние стенки; этот шаг легко перепутать с простым вычитанием.
- Не вычитают общую грань при склейке тел или вычитают её лишь один раз вместо двух.
- Для куба берут 4 грани вместо 6.
Разобранные примеры
Пример 1
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые).
Решение
Площадь поверхности этого многогранника такая же, как площадь поверхности «целого» прямоугольного параллелепипеда, она равна удвоенной сумме попарных произведений его измерений: 2 · (3 · 5 + 4 · 5 + 4 · 3) = 94.
Ответ: 94.
Пример 2
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Найдите площадь поверхности пространственного креста, изображенного на рисунке и составленного из единичных кубов.
Решение
Поверхности креста составлена из шести поверхностей кубов, у каждого из которых отсутствует одна грань. Таким образом, поверхность креста состоит из
Ответ: 30.
Пример 3
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Найдите площадь поверхности многогранника, изображенного на рисунке, все двугранные углы которого прямые.
Решение
Площадь поверхности заданного многогранника складывается из четырех площадей квадратов со стороной 1, двух прямоугольников со сторонами
Ответ: 14.
- «Площадь поверхности составного многогранника», задания ЕГЭ, Образавр (obrazavr.ru)
- «ЕГЭ Профиль. Площадь поверхности и объём составного многогранника. Задача 3», math100.ru
- «Многогранники», задание 3 ЕГЭ, examer.ru
- «Стереометрия», материалы подготовки к ЕГЭ, abitur.ivanovo.ac.ru, 2022
- «Многогранники. Тела вращения», методическая разработка, mpk.brstu.ru, 2021
- У1
Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У2
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У3
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У4
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У5
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У6
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Найдите площадь поверхности многогранника, изображённого на рисунке (все двугранные углы прямые). - У7
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У8
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У9
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У10
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У11
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У12
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0.0,-0.6 z' /%3E%3Cpath d='m 85.2,131.6 c 0,0.8 0,1.6 0,2.5 -1.3,0.2 -0.6,-1.5 -0.7,-2.3 0,-1.4 0,-2.8 0,-4.2 0.8,-0.4 0.8,1.3 1.4,0.0 2.2,-1.0 3.7,2.2 2.4,3.9 -0.6,1.0 -2.4,1.2 -3.1,0.0 z m 2.6,-1.6 c 0.5,-2.5 -3.7,-1.9 -2.6,0.4 0.0,2.1 3.1,1.6 2.6,-0.4 z' /%3E%3Cpath d='m 90.5,129.9 c -0.8,1.9 2.7,2.3 2.0,0.1 0.6,-2.4 -2.3,-2.7 -2.0,-0.1 z m 2.0,4.2 c 0,-0.8 0,-1.6 0,-2.4 -1.8,2.2 -3.8,-1.7 -2.6,-3.3 0.4,-1.8 3.0,0.3 2.6,-1.1 -0.4,-1.1 0.9,-1.8 0.8,-0.3 -0.7,2.2 2.1,-0.6 2.4,1.4 1.2,1.6 -0.3,5.2 -2.4,3.5 -0.4,0.6 0.6,2.5 -0.8,2.1 z m 2.8,-4.2 c 0.8,-1.9 -2.7,-2.3 -2.0,-0.1 -0.6,2.4 2.3,2.7 2.0,0.1 z' /%3E%3C/g%3E%3C/svg%3E)
Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У13
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У14
Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые).
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У16
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Найдите площадь поверхности многогранника, изображённого на рисунке (все двугранные углы прямые). - У17
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У18
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У19
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Найдите площадь поверхности многогранника, изображённого на рисунке (все двугранные углы прямые). - У20
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Найдите площадь поверхности многогранника, изображённого на рисунке (все двугранные углы прямые). - У21
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Найдите площадь поверхности многогранника, изображённого на рисунке. Все двугранные углы многогранника прямые. - У22
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У23
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - У24
Найдите площадь поверхности многогранника, изображенного на рисунке, все двугранные углы которого прямые.
- У25
Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые).
- Д1
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д2
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д3
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д4
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д5
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д6
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Найдите площадь поверхности многогранника, изображенного на рисунке. Все двугранные углы многогранника прямые. - Д7
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д8
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д9
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/%3E%3Cpath d='m 110.7,135.5 c 0.3,1.4 -3.0,1.7 -1.7,0.7 1.5,-0.2 1.0,-1.9 0.5,-2.8 -0.4,-1.0 -0.8,-2.0 -1.2,-3.1 1.2,-0.2 1.2,1.4 1.7,2.2 0.2,0.2 0.4,1.9 0.7,1.2 0.4,-1.1 0.9,-2.2 1.3,-3.4 1.4,-0.2 0.0,1.5 -0.0,2.3 -0.3,0.9 -0.7,1.9 -1.1,2.9 z' /%3E%3Cpath d='m 118.0,132.4 c -0.4,0.8 -1.9,0.1 -2.8,0.3 -2.0,-0.1 0.3,2.4 1.5,1.5 2.5,0.0 -1.4,1.6 -2.3,0.2 -1.9,-1.5 -0.3,-5.3 2.1,-4.4 0.9,0.2 1.4,1.2 1.4,2.2 z m -0.7,-0.2 c 0.6,-2.3 -4.0,-1.0 -2.1,0.0 0.7,0 1.4,-0.0 2.1,-0.0 z' /%3E%3Cpath d='m 119.3,135.0 c 0,-1.5 0,-3.1 0,-4.7 1.1,0 2.2,0 3.3,0 0.0,1.5 -3.3,-0.4 -2.5,1.8 0,0.9 0,1.9 0,2.9 -0.2,0 -0.5,0 -0.7,0 z' /%3E%3Cpath d='m 123.5,134.1 c 0.7,1.4 4.2,-0.8 2.1,-1.2 -0.5,-0.1 -2.2,0.3 -1.5,-0.6 1.0,0.1 3.1,0.0 1.4,-1.2 -0.6,-0.7 -2.8,0.8 -1.8,-0.6 2.0,-0.9 4.2,1.2 3.3,3.2 -0.2,1.5 -3.5,2.3 -3.6,0.7 l 0,-0.1 0,-0.0 0,0 z' /%3E%3Cpath d='m 128.8,133.9 c 1.6,-0.5 0.6,2.3 -0.0,0.6 -0.0,-0.2 0.0,-0.4 0.0,-0.6 z' /%3E%3Cpath d='m 132.2,134.3 c 0,0.8 0,1.6 0,2.5 -1.3,0.2 -0.6,-1.5 -0.7,-2.3 0,-1.4 0,-2.8 0,-4.2 0.8,-0.4 0.8,1.3 1.4,0.0 2.2,-1.0 3.7,2.2 2.4,3.9 -0.6,1.0 -2.4,1.2 -3.1,0.0 z m 2.6,-1.6 c 0.5,-2.5 -3.7,-1.9 -2.6,0.4 0.0,2.1 3.1,1.6 2.6,-0.4 z' /%3E%3Cpath d='m 137.5,132.6 c -0.8,1.9 2.7,2.3 2.0,0.1 0.6,-2.4 -2.3,-2.7 -2.0,-0.1 z m 2.0,4.2 c 0,-0.8 0,-1.6 0,-2.4 -1.8,2.2 -3.8,-1.7 -2.6,-3.3 0.4,-1.8 3.0,0.3 2.6,-1.1 -0.4,-1.1 0.9,-1.8 0.8,-0.3 -0.7,2.2 2.1,-0.6 2.4,1.4 1.2,1.6 -0.3,5.2 -2.4,3.5 -0.4,0.6 0.6,2.5 -0.8,2.1 z m 2.8,-4.2 c 0.8,-1.9 -2.7,-2.3 -2.0,-0.1 -0.6,2.4 2.3,2.7 2.0,0.1 z' /%3E%3C/g%3E%3C/svg%3E)
Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д10
'%3E%3Cpath fill='%230D0F0F' d='M137.616,27.507l2.306-1.114h0.229v7.928c0,0.526,0.022,0.854,0.066,0.982 c0.045,0.13,0.137,0.229,0.276,0.299c0.139,0.068,0.424,0.107,0.853,0.117v0.257h-3.563v-0.257 c0.446-0.009,0.735-0.047,0.866-0.113c0.13-0.067,0.222-0.157,0.272-0.271c0.051-0.112,0.077-0.451,0.077-1.014v-5.069 c0-0.683-0.023-1.122-0.07-1.314c-0.032-0.147-0.092-0.256-0.178-0.325c-0.087-0.069-0.19-0.104-0.313-0.104 c-0.172,0-0.412,0.072-0.72,0.215L137.616,27.507z'/%3E%3C/g%3E%3C/g%3E%3C/g%3E%3Cg%3E%3Cg%3E%3Cdefs%3E%3Crect id='SVGID_3_' x='91.67' y='0.078' width='11.775' height='16.771'/%3E%3C/defs%3E%3CclipPath id='SVGID_4_'%3E%3Cuse xlink:href='%23SVGID_3_' overflow='visible'/%3E%3C/clipPath%3E%3Cg clip-path='url(%23SVGID_4_)'%3E%3Cpath fill='%230D0F0F' d='M94.357,5.364c0.272-0.633,0.615-1.121,1.03-1.465c0.414-0.344,0.931-0.517,1.55-0.517 c0.764,0,1.35,0.245,1.757,0.734c0.31,0.365,0.465,0.755,0.465,1.171c0,0.685-0.438,1.392-1.309,2.12 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id='SVGID_6_'%3E%3Cuse xlink:href='%23SVGID_5_' overflow='visible'/%3E%3C/clipPath%3E%3Cg clip-path='url(%23SVGID_6_)'%3E%3Cpath fill='%230D0F0F' d='M36.98,24.979v0.982h-1.277v2.479h-1.16v-2.479h-4.033v-0.886l4.42-6.218h0.773v6.121H36.98z M34.542,24.979v-4.66l-3.346,4.66H34.542z'/%3E%3C/g%3E%3C/g%3E%3C/g%3E%3Cg%3E%3Cg%3E%3Cdefs%3E%3Crect id='SVGID_7_' x='0' y='104.272' width='11.774' height='16.771'/%3E%3C/defs%3E%3CclipPath id='SVGID_8_'%3E%3Cuse xlink:href='%23SVGID_7_' overflow='visible'/%3E%3C/clipPath%3E%3Cg clip-path='url(%23SVGID_8_)'%3E%3Cpath fill='%230D0F0F' d='M8.622,107.586v0.256c-0.619,0.062-1.124,0.183-1.516,0.364c-0.392,0.183-0.777,0.46-1.16,0.834 c-0.382,0.375-0.697,0.792-0.947,1.251c-0.251,0.46-0.461,1.006-0.629,1.64c0.675-0.457,1.352-0.687,2.03-0.687 c0.65,0,1.216,0.259,1.693,0.775c0.478,0.518,0.717,1.183,0.717,1.996c0,0.784-0.241,1.501-0.724,2.146 c-0.581,0.785-1.351,1.178-2.307,1.178c-0.65,0-1.203-0.212-1.657-0.637c-0.892-0.827-1.335-1.897-1.335-3.216 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style='stroke:none;fill:%23000;fill-opacity:0.4' %3E%3Cpath d='m 103.1,173.3 c 0,0.8 0,1.6 0,2.5 -1.3,0.2 -0.6,-1.5 -0.7,-2.3 0,-1.4 0,-2.8 0,-4.2 0.8,-0.4 0.8,1.3 1.4,0.0 2.2,-1.0 3.7,2.2 2.4,3.9 -0.6,1.0 -2.4,1.2 -3.1,0.0 z m 2.6,-1.6 c 0.5,-2.5 -3.7,-1.9 -2.6,0.4 0.0,2.1 3.1,1.6 2.6,-0.4 z' /%3E%3Cpath d='m 112.0,171.4 c -0.4,0.8 -1.9,0.1 -2.8,0.3 -2.0,-0.1 0.3,2.4 1.5,1.5 2.5,0.0 -1.4,1.6 -2.3,0.2 -1.9,-1.5 -0.3,-5.3 2.1,-4.4 0.9,0.2 1.4,1.2 1.4,2.2 z m -0.7,-0.2 c 0.6,-2.3 -4.0,-1.0 -2.1,0.0 0.7,0 1.4,-0.0 2.1,-0.0 z' /%3E%3Cpath d='m 116.9,173.4 c 0.6,0 1.3,0 2.0,0 0,-1.3 0,-2.7 0,-4.1 1.3,-0.2 0.6,1.5 0.7,2.3 0,0.8 0,1.6 0,2.4 -2.1,0 -4.2,0 -6.4,0 0,-1.5 0,-3.1 0,-4.7 1.3,-0.2 0.6,1.5 0.7,2.3 0.1,0.5 -0.2,1.6 0.2,1.7 0.5,-0.1 1.6,0.2 1.8,-0.2 0,-1.3 0,-2.6 0,-3.9 1.3,-0.2 0.6,1.5 0.7,2.3 0,0.5 0,1.1 0,1.7 z' /%3E%3Cpath d='m 123.3,174.5 c 0.3,1.4 -3.0,1.7 -1.7,0.7 1.5,-0.2 1.0,-1.9 0.5,-2.8 -0.4,-1.0 -0.8,-2.0 -1.2,-3.1 1.2,-0.2 1.2,1.4 1.7,2.2 0.2,0.2 0.4,1.9 0.7,1.2 0.4,-1.1 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/%3E%3Cpath d='m 150.1,171.6 c -0.8,1.9 2.7,2.3 2.0,0.1 0.6,-2.4 -2.3,-2.7 -2.0,-0.1 z m 2.0,4.2 c 0,-0.8 0,-1.6 0,-2.4 -1.8,2.2 -3.8,-1.7 -2.6,-3.3 0.4,-1.8 3.0,0.3 2.6,-1.1 -0.4,-1.1 0.9,-1.8 0.8,-0.3 -0.7,2.2 2.1,-0.6 2.4,1.4 1.2,1.6 -0.3,5.2 -2.4,3.5 -0.4,0.6 0.6,2.5 -0.8,2.1 z m 2.8,-4.2 c 0.8,-1.9 -2.7,-2.3 -2.0,-0.1 -0.6,2.4 2.3,2.7 2.0,0.1 z' /%3E%3C/g%3E%3C/svg%3E)
Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д11
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д12
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Найдите площадь поверхности многогранника, изображённого на рисунке (все двугранные углы прямые). - Д13
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style='stroke:none;fill:%23000;fill-opacity:0.4' %3E%3Cpath d='m 89.3,115.5 c 0,0.8 0,1.6 0,2.5 -1.3,0.2 -0.6,-1.5 -0.7,-2.3 0,-1.4 0,-2.8 0,-4.2 0.8,-0.4 0.8,1.3 1.4,0.0 2.2,-1.0 3.7,2.2 2.4,3.9 -0.6,1.0 -2.4,1.2 -3.1,0.0 z m 2.6,-1.6 c 0.5,-2.5 -3.7,-1.9 -2.6,0.4 0.0,2.1 3.1,1.6 2.6,-0.4 z' /%3E%3Cpath d='m 98.2,113.6 c -0.4,0.8 -1.9,0.1 -2.8,0.3 -2.0,-0.1 0.3,2.4 1.5,1.5 2.5,0.0 -1.4,1.6 -2.3,0.2 -1.9,-1.5 -0.3,-5.3 2.1,-4.4 0.9,0.2 1.4,1.2 1.4,2.2 z m -0.7,-0.2 c 0.6,-2.3 -4.0,-1.0 -2.1,0.0 0.7,0 1.4,-0.0 2.1,-0.0 z' /%3E%3Cpath d='m 103.1,115.6 c 0.6,0 1.3,0 2.0,0 0,-1.3 0,-2.7 0,-4.1 1.3,-0.2 0.6,1.5 0.7,2.3 0,0.8 0,1.6 0,2.4 -2.1,0 -4.2,0 -6.4,0 0,-1.5 0,-3.1 0,-4.7 1.3,-0.2 0.6,1.5 0.7,2.3 0.1,0.5 -0.2,1.6 0.2,1.7 0.5,-0.1 1.6,0.2 1.8,-0.2 0,-1.3 0,-2.6 0,-3.9 1.3,-0.2 0.6,1.5 0.7,2.3 0,0.5 0,1.1 0,1.7 z' /%3E%3Cpath d='m 109.5,116.7 c 0.3,1.4 -3.0,1.7 -1.7,0.7 1.5,-0.2 1.0,-1.9 0.5,-2.8 -0.4,-1.0 -0.8,-2.0 -1.2,-3.1 1.2,-0.2 1.2,1.4 1.7,2.2 0.2,0.2 0.4,1.9 0.7,1.2 0.4,-1.1 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Найдите площадь поверхности многогранника, изображённого на рисунке (все двугранные углы прямые). - Д14
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д15
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д16
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Найдите площадь поверхности многогранника, изображённого на рисунке (все двугранные углы прямые). - Д17
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д18
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Найдите площадь поверхности многогранника, изображённого на рисунке (все двугранные углы прямые). - Д19
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Найдите площадь поверхности многогранника, изображённого на рисунке (все двугранные углы прямые). - Д20
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Найдите площадь поверхности многогранника, изображённого на рисунке (все двугранные углы прямые). - Д21Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые).
- Д22
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые). - Д23
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Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые).
- Д24
Найдите площадь поверхности многогранника, изображенного на рисунке (все двугранные углы прямые).
